An Anosov diffeomorphism is a diffeomorphism of a manifold to itself such that the tangent bundle of is hyperbolic with respect to . Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's cat map.
A hyperbolic linear map with integer entries in the transformation matrix and determinant is an Anosov diffeomorphism of the -torus. Not every manifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are expansive, and there are no Anosov diffeomorphisms on the circle.
It is conjectured that if is an Anosov diffeomorphism on a compact Riemannian manifold and the nonwandering set of is , then is topologically conjugate to a finite-to-one factor of an Anosov automorphism of a nilmanifold. It has been proved that any Anosov diffeomorphism on the -torus is topologically conjugate to an Anosov automorphism, and also that Anosov diffeomorphisms are structurally stable.